Research Article
Benjamin Cooper
Categorical representations of the modular group
Abstract: Actions of the modular group on categories are constructed. A hyperelliptic involution is used to
convert the braid representations underlying Khovanov homology to representations of the modular group.
Keywords: Knot homology, mapping class groups, braid groups MSC 2010: 20F36, 18G35
||
Benjamin Cooper:Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland, e-mail: benjamin.cooper@math.uzh.ch
Communicated by: Frederick R. Cohen
1 Introduction
The study of group actions on categories is a rich topic which arises in many areas of mathematics. Such group actions appear when topological field theories are present. An𝑛-dimensional topological field theory Z associates to each(𝑛 − 2)-manifold Σ a category Z(Σ) upon which the mapping class group Γ(Σ). In particular, there are equivalences
𝐹𝑔: Z(Σ) → Z(Σ) for each 𝑔 ∈ Γ(Σ).
When𝑛 = 4, one expects many examples of mapping class groups Γ(Σ) acting on categories Z(Σ), see [14, 19]. While there are examples of categorical braid group actions in the literature [15, 20, 24, 27], there are very few examples for surfaces of genus greater than zero; although, see [16].
In this paper we present a means by which the categories underlying the Khovanov homology of knots and links can be used to produce categories which are representations of the modular groupSL(2, ℤ) (the mapping class group of the torus). A hyperelliptic involution is used to reduce the mapping class group of the torus to the mapping class group of the four punctured sphere. This identification allows us to construct several families of modular group representations from braid group representations.
The theory of modular functors shows that it is possible to construct representations for surfaces of arbitrary genus when given a family of braid group representations and a representation of the modular group satisfying some compatibility conditions [1, 18, 26]. In this sense, the special case ofSL(2, ℤ) actions is an important component of the most general case.
Organization
In Section 2, definitions related to knot homology and group actions are recalled. In Section 3, we review information about braid groups and mapping class groups. In Section 4, we construct families of cate-goriesR𝑛,𝑘andK𝑛which supportSL(2, ℤ) actions.
2 Algebraic background
In this section we collect some algebraic background information. We begin by recalling the differential graded categoriesKom∗1/2(𝑛) and the existence of certain chain complexes 𝑃𝑛,𝑘withinKom∗1/2(𝑛). The defi-nition of a group action on a category is given and a construction for reducing the gradings is introduced.
2.1 Cobordisms
Dror Bar-Natan’s graphical formulation [2] of the Khovanov categorification [14] consists of a series of cate-gories:Cob(𝑛) and Kom(𝑛). The construction in Section 4 will take place within categories related to Kom(𝑛). Definitions are reviewed below. They are the same as those appearing in [7, 8] with the exception that we adopt the Kauffman bracket grading convention from Rozansky [23].
2.1.1 Categories of cobordisms
Let𝑅 be a field.
Definition 2.1. The category of sl2-cobordisms with2𝑛 boundary points will be denoted by Cob(𝑛).
In more detail, there is a pre-additive categoryPre-Cob(𝑛) whose objects are formally ℤ-graded diagrams. Each diagram is given by an isotopy class of1-manifolds embedded in a disk 𝐷2relative to2𝑛 boundary points. There is a functor
𝑞 : Pre-Cob(𝑛) → Pre-Cob(𝑛)
which increases the grading by one. The morphisms of the categoryPre-Cob(𝑛) are given by 𝑅-linear combi-nations of isotopy classes of orientable cobordisms embedded in𝐷2× 𝐼, see [2].
The internal degree of a cobordism𝐶 : 𝑞𝑖𝐴 → 𝑞𝑗𝐵 is the sum of its topological degree and its 𝑞-degree: deg(𝐶) = deg𝜒(𝐶) + deg𝑞(𝐶).
The topological degree
deg𝜒(𝐶) = 𝜒(𝐶) − 𝑛
is given by the Euler characteristic of𝐶 and the 𝑞-degree deg𝑞(𝐶) = 𝑗 − 𝑖
is given by the relative difference in𝑞-gradings.
We impose the relations implied by the requirement that the circle is isomorphic to the direct sum of two empty sets,
≅ 𝑞⌀ ⊕ 𝑞−1⌀, (2.1)
using maps of internal degree zero (see [3]). In doing so, we obtain the categoryCob(𝑛) as a quotient of the additive closureMat(Pre-Cob(𝑛)) of Pre-Cob(𝑛).
There is a composition
⊗ : Cob(𝑛) × Cob(𝑛) → Cob(𝑛).
It is defined by gluing diagrams and cobordisms. The unit1𝑛with respect to this composition is a diagram consisting of𝑛 parallel lines.
2.1.2 Categories of half-graded chain complexes
Definition 2.2. IfA is an additive category, then Kom∗1/2(A) will denote the differential graded category of
1
2-graded chain complexes inA.
In more detail, the objects ofKom∗1/2(A) are chain complexes 𝐶 = (𝐶𝑖, 𝑑𝐶𝑖)𝑖∈1
2ℤwhere𝐶𝑖∈ A and 𝑑
𝐶
𝑖 : 𝐶𝑖→ 𝐶𝑖+1.
Each chain complex𝐶 is bounded from below: 𝐶𝑖= 0 for 𝑖 sufficiently small. Suppose that𝐶 = (𝐶𝑖, 𝑑𝐶𝑖)𝑖∈1
2ℤand𝐷 = (𝐷𝑖, 𝑑
𝐷
𝑖 )𝑖∈12ℤare two chain complexes inKom ∗
1/2(A). Then a map
𝑔 : 𝐶 → 𝐷 of 𝑡-degree 𝑙 is a collection
of maps inA. The vector space of maps with 𝑡-degree 𝑙 will be denoted by Hom𝑙(𝐶, 𝐷). There is a composition ∘ : Hom𝑙(𝐶, 𝐷) ⊗ Hom𝑚(𝐷, 𝐸) → Hom𝑙+𝑚(𝐶, 𝐸)
and there are identity maps1𝐶= {1𝐶𝑖 : 𝐶𝑖→ 𝐶𝑖} of degree zero.
There is a differential𝑑 : Hom𝑙(𝐶, 𝐷) → Hom𝑙+1(𝐶, 𝐷) which is defined on a map 𝑔 = {𝑔𝑖} of 𝑡-degree 𝑙 by the formula
(𝑑𝑔)𝑖+1= 𝑔𝑖+1∘ 𝑑𝐶𝑖 + (−1) 𝑙+1𝑑𝐷
𝑖+𝑙∘ 𝑔𝑖. (2.2)
If𝑔 ∈ Hom𝑙(𝐶, 𝐷) and ℎ ∈ Hom𝑚(𝐶, 𝐷) are maps of 𝑡-degree 𝑙 and 𝑚 respectively, then the differential 𝑑 satis-fies the Leibniz rule
𝑑(ℎ ∘ 𝑔) = ℎ ∘ (𝑑𝑔) + (−1)𝑙(𝑑ℎ) ∘ 𝑔.
If 1𝐶: 𝐶 → 𝐶 is the identity map, then 𝑑(1𝐶) = 0. Let Hom∗(𝐶, 𝐷) = ∏𝑙Hom𝑙(𝐶, 𝐷) be the maps from 𝐶 to𝐷. Then the pairs (Hom∗(𝐶, 𝐷), 𝑑) are chain complexes and the category Kom∗1/2(A) is a differential graded category [25].
Definition 2.3. The homotopy categoryHo1/2(A) of Kom∗1/2(A) has the same objects as the category Kom∗1/2(A).
The maps inHo1/2(A) are given by the zeroth homology groups of the maps in Kom∗1/2(A): HomHo1/2(A)(𝐶, 𝐷) = H
0(Hom∗(𝐶, 𝐷)),
see [25, Definition 1].
Two objects𝐶 and 𝐷 in Kom∗1/2(A) are homotopic, 𝐶 ≃ 𝐷, when they are isomorphic in the homotopy categoryHo1/2(A). A chain complex 𝐶 is contractible when 𝐶 ≃ 0.
Definition 2.4. The category of12-graded chain complexes of cobordismsKom∗1/2(𝑛) is the differential graded
category of1
2-graded chain complexes inCob(𝑛):
Kom∗
1/2(𝑛) = Kom ∗
1/2(Cob(𝑛)).
2.1.3 Tangle invariants
Definition 2.5. The skein relation associated to a crossing is the cone complex
= 𝑡12 → 𝑡−12 . (2.3)
The definition above allows one to associate chain complexes in the categoryKom∗1/2(𝑛) to diagrams of framed(𝑛, 𝑛)-tangles. Any two such diagrams which differ by Reidemeister moves 2 or 3 are assigned to homotopy equivalent chain complexes. The first Reidemeister move results in a degree shift, see Property 2 in Section 2.2.1.
2.2 Projectors
In [8], V. Krushkal and the author defined a special chain complex𝑃𝑛. This construction is summarized by the theorem below. See also [12, 21].
Theorem 2.6. There exists a chain complex𝑃𝑛∈ Kom∗
1/2(𝑛) called the universal projector which satisfies:
(1) 𝑃𝑛is positively graded with differential having internal degree zero.
(2) The complex𝑃𝑛⊗ 𝐷 is contractible for any diagram 𝐷 which is not the identity 1𝑛.
(3) The identity object appears only in homological degree zero and only once.
In [7], M. Hogancamp and the author constructed a series of projectors𝑃𝑛,𝑘∈ Kom∗1/2(𝑛) where 𝑘 = 𝑛, 𝑛 − 2, 𝑛 − 4, . . . , 𝑛 (mod 2). (The bottom projector 𝑃𝑛,𝑛 (mod 2)is𝑃𝑛,0when𝑛 is even and 𝑃𝑛,1when𝑛 is odd.) These higher
order projectors are characterized by properties analogous to those mentioned in the theorem above.
Definition 2.7. Suppose that𝐶 = (𝐶𝑖, 𝑑𝑖𝐶)𝑖∈1
2ℤis a chain complex in Kom
∗
1/2(𝑛) and let 𝐶𝑖= ⨁𝑗𝐶𝑖,𝑗so that
each object 𝐶𝑖,𝑗∈ Pre-Cob(𝑛) is a diagram. Then the through-degree 𝜏(𝐶) is the maximum over all of the diagrams𝐶𝑖,𝑗of the minimum number of vertical strands in any cross section of𝐶𝑖,𝑗.
In other words,𝜏(𝐶) is the largest number of lines connecting the bottom to the top of any diagram 𝐶𝑖,𝑗in𝐶. Since vertical lines can only be added or removed in pairs, we have
𝜏(𝐶) ∈ {𝑛, 𝑛 − 2, 𝑛 − 4, . . . , 𝑛 (mod 2)}.
Theorem 2.8. For each positive integer𝑛, there is a series of higher order projectors 𝑃𝑛,𝑘∈ Kom∗
1/2(𝑛) where
𝑘 = 𝑛, 𝑛 − 2, 𝑛 − 4, . . . , 𝑛 (mod 2) which satisfy the properties: (1) The𝑃𝑛,𝑘have through-degree𝜏(𝑃𝑛,𝑘) = 𝑘.
(2) If𝐷 is diagram with through-degree 𝜏(𝐷) < 𝑘, then
𝐷 ⊗ 𝑃𝑛,𝑘≃ 0 and 𝑃𝑛,𝑘⊗ 𝐷 ≃ 0.
Together with a normalization axiom, analogous to (3) in Theorem 2.6 above, these properties characterize each object𝑃𝑛,𝑘uniquely up to homotopy.
The higher order projectors satisfy orthogonality and idempotence relations. They also fit together to form a convolution chain complex that is homotopy equivalent to the identity object ([7, Section 8]):
𝑃𝑛,𝑙⊗ 𝑃𝑛,𝑘≃ 𝛿𝑙𝑘𝑃𝑛,𝑘 and 1𝑛≃ 𝑃𝑛,𝑛 (mod 2)→ ⋅ ⋅ ⋅ → 𝑃𝑛,𝑛−4→ 𝑃𝑛,𝑛−2→ 𝑃𝑛,𝑛. (2.4)
Remark 2.9. The objects𝑃𝑛 and𝑃𝑛,𝑘 were originally defined in categoriesKom(𝑛). By construction, every object ofKom(𝑛) is an object of Kom∗1/2(𝑛). This identification allows us to state the theorems as we have above.
2.2.1 Tangles and twists
If the projector𝑃𝑛,𝑘is represented by a box, then it satisfies the two properties below.
Property 1(Drags through tangles). We have
≃ and ≃ .
Property 2(Absorbs twists). We have
≃ 𝑡12𝑘2𝑞𝑘 and ≃ 𝑡−12𝑘2𝑞−𝑘 .
For the degree shifts above to be accurate, the skein relation must be given by equation (2.3). When𝑛 = 1 and𝑘 = 1, Property 2 determines the grading shift associated to the first Reidemeister move.
2.3 Group actions on categories
Definition 2.10. If𝐺 is a group and C is a differential graded category, then an action of 𝐺 on C is a
homomor-phism from the group𝐺 to the category End(C) of functors from C to C: 𝐹 : 𝐺 → End(C) such that 𝐹(𝑔ℎ) ≃ 𝐹(𝑔) ∘ 𝐹(ℎ). For discussion of dg functors see [25, Section 2.3].
If𝐺 = ⟨𝑆 : 𝑅⟩ is a presentation for the group 𝐺, then a group action is obtained by specifying functors 𝐹𝑠, 𝐹𝑠−1 : C → C for each generator 𝑠 ∈ 𝑆, and homotopy equivalences
𝐹𝑠1∘ 𝐹𝑠2∘ ⋅ ⋅ ⋅ ∘ 𝐹𝑠𝑁≃ 𝐹𝑠1∘ 𝐹𝑠2∘ ⋅ ⋅ ⋅ ∘ 𝐹𝑠𝑀 (2.5)
whenever the words𝑠1𝑠2⋅ ⋅ ⋅ 𝑠𝑁and𝑠1𝑠2⋅ ⋅ ⋅ 𝑠𝑀coincide by virtue of the relations in𝑅. To accomplish this, it suffices to find homotopy equivalences,𝐹𝑠1∘ ⋅ ⋅ ⋅ ∘ 𝐹𝑠𝑁 ≃ IdCfor each𝑟 ∈ 𝑅, 𝑟 = 𝑠1⋅ ⋅ ⋅ 𝑠𝑁.
When a group𝐺 acts on a dg category C, there is an induced action of 𝐺 on the homotopy category Ho(C) ofC. If Ho(C) is a triangulated category, then the action of 𝐺 on Ho(C) agrees with definitions found in the references [15, 20, 24].
The notion of group action that is given above is sometimes called weak. A strong action is one in which the homotopy equivalences (2.5) are uniquely determined. We will only prove that the actions defined here are weak, see Remark 4.7.
In Section 4, we will constructSL(2, ℤ) actions on categories R𝑛,𝑘andK𝑛.
2.4 Reducing the grading
In this subsection we introduce cyclically graded dg categoriesC/𝑡𝑛2𝑞𝑚. IfC is a1
2-graded dg category, such
asKom∗1/2(A), then there are functors
𝑡12, 𝑞 : C → C, (2.6)
which increase the𝑡 and 𝑞 gradings by12and1 respectively. The categories C/𝑡𝑛2𝑞𝑚are obtained by collapsing
the grading so that the identity
𝑡𝑛2𝑞𝑚≅ Id
C/𝑡𝑛2𝑞𝑚
holds in the categoryEnd(C/𝑡𝑛2𝑞𝑚). This is accomplished by extending a technique used to define ℤ/2-graded
dg categories [10, Section 5.1].
Definition 2.11. For each pair of integers,𝑛, 𝑚 ∈ ℤ, there is a 𝑞-graded differential graded algebra (𝐿𝑛,𝑚, 𝑑)
which is given by
𝐿𝑛,𝑚= 𝑅[𝑡
1
2, 𝑞]/(𝑡𝑛2𝑞𝑚= 1) and 𝑑 = 0.
The ring𝑅 is chosen to be the ground field of the category C. The grading of 𝐿𝑛,𝑚is determined by the table below:
𝑞: deg𝑞= 1, deg𝑡= 0,
𝑡12: deg
𝑞= 0, deg𝑡= 12.
Definition 2.12. LetC be a 12-graded dg category. An 𝐿𝑛,𝑚-module in C is a pair (𝐶, 𝑓) consisting of an
object𝐶 ∈ C and a map of graded dg algebras
𝑓 : 𝐿𝑛,𝑚→ End(𝐶).
The categoryC/𝑡𝑛2𝑞𝑚of𝐿
𝑛,𝑚-modules inC is the dg category consisting of 𝐿𝑛,𝑚-modules and𝐿𝑛,𝑚-equivariant
maps.
In more detail, the objects ofC/𝑡𝑛2𝑞𝑚are pairs(𝐶, 𝑓) consisting of objects 𝐶 ∈ C and maps 𝑓 : 𝐿
𝑛,𝑚→ End(𝐶)
of differential graded algebras which preserve the gradings. A mapℎ : (𝐶, 𝑓) → (𝐷, 𝑔) of degree 𝑙 is a map ℎ : 𝐶 → 𝐷 of degree 𝑙 in C which commutes with the action of 𝐿𝑛,𝑚on𝐶 and 𝐷 respectively:
𝐶 𝐷 𝐶 𝐷 ← ℎ→ ←→ 𝑓(𝑎) ←→𝑔(𝑎) ← ℎ→ (2.7) for all𝑎 ∈ 𝐿𝑛,𝑚.
The functors𝑡12, 𝑞 : C → C from equation (2.6) induce functors 𝑡12, 𝑞 : C/𝑡2𝑛𝑞𝑚 → C/𝑡𝑛2𝑞𝑚. For example,
if(𝐶, 𝑓) is an object of C/𝑡𝑛2𝑞𝑚, then
𝑡𝑘2𝑞𝑙(𝐶, 𝑓) = (𝑡𝑘2𝑞𝑙𝐶, 𝜑 ∘ 𝑓)
where𝜑 is the natural isomorphism End(𝐶) → End(𝑡𝑘2𝑞𝑙𝐶).
The proposition below shows that the equation𝑡𝑛2𝑞𝑚≅ Id
C/𝑡𝑛2𝑞𝑚holds inEnd(C/𝑡
𝑛 2𝑞𝑚).
Proposition 2.13. Suppose that𝑡12, 𝑞 : C/𝑡𝑛2𝑞𝑚→ C/𝑡𝑛2𝑞𝑚denote the functors which increase the𝑡-degree and
𝑞-degree in C/𝑡𝑛2𝑞𝑚by1
2and1 respectively. Then there is an isomorphism
𝜂 : 𝑡𝑛2𝑞𝑚≅ Id
C/𝑡𝑛2𝑞𝑚
in the functor categoryEnd(C/𝑡𝑛2𝑞𝑚).
Proof. The definition ofC/𝑡𝑛2𝑞𝑚 implies that for each object(𝐶, 𝑓) ∈ C/𝑡𝑛2𝑞𝑚there are maps 𝑓(𝑡12) : 𝐶 → 𝐶
and 𝑓(𝑞) : 𝐶 → 𝐶. These maps determine isomorphisms: 𝜂𝐶: 𝑡𝑛2𝑞𝑚(𝐶, 𝑓) → (𝐶, 𝑓). If ℎ : (𝐶, 𝑓) → (𝐷, 𝑔) is
a map inC/𝑡𝑛2𝑞𝑚, then equation (2.7) implies thatℎ commutes with 𝜂
𝐶. So the collection𝜂 = {𝜂𝐶} is a natural
transformation𝑡𝑛2𝑞𝑚→ Id. The map 𝜂 is the isomorphism in the statement above.
Suppose thatgVect𝑅is the category of graded vector spaces over𝑅. Then the example below illustrates the material introduced above.
Example 2.14. An𝐿0,𝑚-module in the dg categoryKom∗1/2(gVect𝑅) is a chain complex 𝐶 on which the grading
shift functor𝑞𝑚: 𝐶 → 𝐶 acts by identity. Similarly, an 𝐿𝑛,0-module𝐶 in the category Kom∗1/2(gVect𝑅) is a chain complex on which the functor𝑡𝑛2 : 𝐶 → 𝐶 acts by identity.
The categoriesKom∗1/2(gVect𝑅)/𝑡0𝑞𝑚 andKom∗1/2(gVect𝑅)/𝑡𝑛2𝑞0 consist of𝑞𝑚-cyclic and𝑡𝑛2-cyclic chain
complexes respectively. There is a forgetful functor
𝑈 : C/𝑡𝑛2𝑞𝑚→ C
which is determined by the assignment(𝐶, 𝑓) → 𝐶. The functor 𝑈 has a left adjoint 𝑃 : C → C/𝑡𝑛2𝑞𝑚which
sends a chain complex𝐶 to 𝑃(𝐶) = 𝐿𝑛,𝑚⊗𝑅𝐶. If 𝜋 : 1
2ℤ × ℤ → 1
2ℤ × ℤ/(𝑛, 𝑚) is the quotient map, then
𝑃(𝐶)𝑖,𝑗= ⨁ (𝑘, 𝑙)∈𝜋−1(𝑖,𝑗)
𝐶𝑘,𝑙
where𝐶𝑘,𝑙denotes the value of𝐶 in 𝑡-degree 𝑘 and 𝑞-degree 𝑙. The proposition below records this information.
Proposition 2.15. There is an adjunction
𝑃 : C C/𝑡𝑛2𝑞𝑚: 𝑈.
The proof that the free𝐿𝑛,𝑚-module functor𝑃 = 𝐿𝑛,𝑚⊗𝑅− is left adjoint to the forgetful functor 𝑈 is standard, see [17, Section IV.2].
In the remainder of this section we specialize to chain complexes.
Remark 2.16. If𝑈(𝐶) and 𝑈(𝐷) are cyclically graded complexes, then 𝑈(𝐶) ⊗ 𝑈(𝐷) is a cyclically graded
complex. IfA is a monoidal category, then Kom∗1/2(A)/𝑡𝑛2𝑞𝑚is a monoidal category.
Proposition 2.17. The homotopy categoryHo1/2(A)/𝑡𝑛2𝑞𝑚ofKom∗
1/2(A)/𝑡
𝑛
2𝑞𝑚is a triangulated category.
The proof assumes familiarity with the reference [5].
Proof. The functors𝑃 and 𝑈 commute with totalization.
In more detail, letC = Kom∗1/2(A)/𝑡𝑛2𝑞𝑚. Given a twisted complex𝐸 = {𝐸
𝑖, 𝑟𝑖𝑗} over C ([5, Section 1,
Defini-tion 1]). Let𝛼(𝐸) : Cop → Kom be the functor determined by 𝐸 ([5, Section 1, Definition 3]). By definition, 𝛼(𝐸)(𝑋) = ⨁
𝑖
𝑡𝑖Hom C(𝑋, 𝐸𝑖).
The differential of this chain complex is the sum𝑑 + 𝑄 where 𝑑 is the Hom-differential from equation (2.2) and𝑄 = (𝑟𝑖𝑗).
Let𝑇(𝐸) = ⨁𝑖𝑡𝑖𝐸𝑖∈ D be the chain complex with differential (𝑟𝑖𝑗). Then the chain complex 𝑇(𝐸) rep-resents the functor𝛼(𝐸). This is because the functor ℎ𝑇(𝐸): Cop → Kom associated to 𝑇(𝐸) by the Yoneda embedding is given on objects𝑋 by
ℎ𝑇(𝐸)(𝑋) = Hom(𝑋, 𝑇(𝐸)) = Hom(𝑋, ⨁ 𝑖 𝑡𝑖𝐸 𝑖) ≅ ⨁ 𝑖 𝑡𝑖Hom(𝑋, 𝐸 𝑖).
Checking that the differentials agree implies thatℎ𝑇(𝐸)(𝑋) ≅ 𝛼(𝐸)(𝑋). This isomorphism does not depend on𝑋, so ℎ𝑇(𝐸)≅ 𝛼(𝐸).
Since each twisted complex 𝐸 is represented by a chain complex 𝑇(𝐸) in C, the category C is pre-triangulated ([5, Section 3, Definition 1]). It follows that the homotopy categoryHo(C) = Ho1/2(A)/𝑡𝑛2𝑞𝑚is
triangulated ([5, Section 3, Proposition 2]). The argument above also applies to subcategories.
Corollary 2.18. Cyclic reductions of pre-triangulated subcategoriesS ⊂ Kom∗1/2(A) have triangulated homotopy
categoriesHo1/2(S)/𝑡𝑛2𝑞𝑚.
The proof of the main theorem shows that the modular group relations hold in a certain categoryE𝑛,𝑘up to a grading shift s𝑘: E𝑛,𝑘→ E𝑛,𝑘. The construction introduced in this section allows us to reduce the grading so that the relations of the modular group hold without the grading shift.
3 Braid groups and the modular group
In this section we recall the relationship between the braid groupB3 and the modular group PSL(2, ℤ). Notation established here will be used later.
3.1 Braid groups
If𝐷𝑛is the closed disk with𝑛 punctures, then the braid group B𝑛on𝑛 strands is given by the path components of the group of boundary and orientation preserving diffeomorphisms of𝐷𝑛:
B𝑛= 𝜋0(Diff+(𝐷𝑛, 𝜕𝐷𝑛)).
Alternatively, suppose thatConfig𝑛(𝐷2) is the configuration space of 𝑛 distinct points in the unit disk. Then the symmetric group𝑆𝑛acts onConfig𝑛(𝐷2) by permuting these points and the braid group B𝑛is the fundamental group of the quotient
B𝑛= 𝜋1(Config𝑛(𝐷 2)/𝑆
𝑛, 𝑥0),
see [4, 11]. Each element[𝛾] ∈ B𝑛determines a loop𝛾 ∈ [𝛾], 𝛾 : (𝐼, 𝜕𝐼) → (Config𝑛(𝐷2)/𝑆𝑛, 𝑥0). Such a map 𝛾 factors as𝛾(𝑡) = (𝛾1(𝑡), . . . , 𝛾𝑛(𝑡)) where 𝛾𝑖: 𝐼 → 𝐷2. The composition of the union∐𝑖𝛾𝑖with the fold map is an embedding ̃𝛾 : ∐𝑖𝐼 → 𝐷2× 𝐼 which determines a braid im( ̃𝛾) in 3-space. Each such braid can be represented by a diagram consisting of a sequence of crossings𝜎𝑖for1 ≤ 𝑖 < 𝑛.
When𝑛 = 3, a presentation for the braid group in terms of these generators is given by B3= ⟨𝜎1, 𝜎2: 𝜎1𝜎2𝜎1= 𝜎2𝜎1𝜎2⟩.
3.2 Modular groups
The modular groupSL(2, ℤ) is the group of 2 × 2 matrices with integer coefficients and determinant 1: SL(2, ℤ) = {(𝑎 𝑏𝑐 𝑑) : 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℤ and 𝑎𝑑 − 𝑏𝑐 = 1}.
It is also the mapping class group of the torus𝑇, see [11]. If 𝐼 ∈ SL(2, ℤ) is the identity matrix, then the center Z(SL(2, ℤ)) ⊂ SL(2, ℤ) is the subgroup {𝐼, −𝐼}. The projective modular group PSL(2, ℤ) is the quotient
PSL(2, ℤ) = SL(2, ℤ)/{𝐼, −𝐼}.
Choosing representatives for the elements{𝐼, −𝐼} yields a ℤ/2-action on the torus with quotient: 𝑆24= 𝑇/(ℤ/2), the four punctured sphere. The groupPSL(2, ℤ) is the mapping class group of 𝑆24. It can be shown thatPSL(2, ℤ) is isomorphic to the free productℤ/2 ∗ ℤ/3; equivalently,
PSL(2, ℤ) ≅ ⟨𝑎, 𝑏 : 𝑎2= 1, 𝑏3= 1⟩.
3.3 Relating
B
3and
PSL(2, ℤ)
Denote by𝑞𝑛the braid
𝑞𝑛= 𝜎𝑛−1𝜎𝑛−2⋅ ⋅ ⋅ 𝜎1∈ B𝑛.
For instance,
𝑞3= .
Multiplying𝑞𝑛by itself𝑛 times produces the full twist, 𝑇𝑛= 𝑞𝑛𝑛. The equation
𝜎𝑖𝑇𝑛 = 𝑇𝑛𝜎𝑖 for1 ≤ 𝑖 < 𝑛 (3.1)
can be seen by dragging𝜎𝑖through from the bottom of𝑇𝑛to the top of𝑇𝑛. It follows that𝑇𝑛is contained in the centerZ(B𝑛) ⊂ B𝑛of the braid group. In fact,⟨𝑇𝑛⟩ = Z(B𝑛), see [6].
The following proposition tells us that the quotientB3/⟨𝑇3⟩ is related to the modular group.
Proposition 3.1. The quotient ofB3by the subgroup generated by the full twist𝑇3is isomorphic to the projective modular groupPSL(2, ℤ),
B3/⟨𝑇3⟩ ≅ PSL(2, ℤ).
Proof. If we set𝑎 = 𝜎1𝜎2𝜎1and𝑏 = 𝜎1𝜎2, then
B3/⟨𝑇3⟩ = ⟨𝜎1, 𝜎2: 𝜎1𝜎2𝜎1= 𝜎2𝜎1𝜎2, 𝑇3= 1⟩ ≅ ⟨𝑎, 𝑏 : 𝑎
2= 1, 𝑏3= 1⟩ ≅ PSL(2, ℤ).
The groupsB𝑛/⟨𝑇𝑛⟩ are always related to mapping class groups of punctured spheres, see [4, Section 4.2].
4 The construction
In this section we construct sequences of categoriesR𝑛,𝑘and prove that each category supports an action of the modular group. We begin by defining categoriesE𝑛,𝑘as extensions ofKom∗1/2(3𝑛) from Section 2.1. The categoriesR𝑛,𝑘are obtained fromE𝑛,𝑘using the construction in Section 2.4. In Theorem 4.5 below, it is shown that there is an action of the groupPSL(2, ℤ) on each category R𝑛,𝑘. This induces an action of the modular group onR𝑛,𝑘.
The first step is to define a chain complex𝐸𝑛,𝑘consisting of a composition of projectors. This object is illustrated in the proof of Theorem 4.5.
Definition 4.1(𝐸𝑛,𝑘). Suppose that 𝑃𝑛,𝑘is the higher order projector from Theorem 2.8. Then set 𝐸𝑛,𝑘= (𝑃𝑛,𝑘⊔ 𝑃𝑛,𝑘⊔ 𝑃𝑛,𝑘) ⊗ 𝑃3𝑛,𝑘.
Definition 4.2(E𝑛,𝑘). The category E𝑛,𝑘is the full dg subcategory ofKom∗1/2(3𝑛) consisting of objects which are homotopy equivalent to those of the form𝐶 ⊗ 𝐸𝑛,𝑘.
Each object in the categoryE𝑛,𝑘is bigraded. One can shift either the𝑡-degree or the 𝑞-degree of each object using the functors𝑡12or𝑞. Property 2 in Section 2.2 suggests the following definition.
Definition 4.3(s𝑘). The shift functor s𝑘: E𝑛,𝑘→ E𝑛,𝑘is the composition s𝑘= 𝑡𝑘2𝑞2𝑘. If𝐶 ∈ E𝑛,𝑘is a chain complex, then
deg𝑡(s𝑘(𝐶)) = deg𝑡(𝐶) + 𝑘
2 and deg
𝑞(s𝑘(𝐶)) = deg𝑞(𝐶) + 2𝑘.
Definition 4.4(R𝑛,𝑘). The category R𝑛,𝑘is obtained using the construction from Section 2.4: R𝑛,𝑘= E𝑛,𝑘/s𝑘.
Theorem 4.5. The groupPSL(2, ℤ) acts on the category R𝑛,𝑘.
Proof. Using the presentationPSL(2, ℤ) = ⟨𝑎, 𝑏 : 𝑎2= 1, 𝑏3= 1⟩ from Section 3.2, we will construct a functor
PSL(2, ℤ) → End(R𝑛,𝑘) where 𝑔 → 𝐹𝑔.
Associated to the generators𝑎 and 𝑏 are the chain complexes that are determined by the tangles
𝑎 = and 𝑏 =
where each line represents an𝑛-cabled strand. There are functors, 𝐹𝑎, 𝐹𝑏: R𝑛,𝑘→ R𝑛,𝑘, given by tensoring on the left. To the identity element1 ∈ PSL(2, ℤ), we associate the functor 𝐹1where1 = 13𝑛is the diagram con-sisting of3𝑛 vertical strands. For inverses, 𝑎−1and𝑏−1, we use the functors𝐹𝑎and𝐹𝑏2. To a word𝑤 ⋅ 𝑤∈ ⟨𝑎, 𝑏⟩,
we associate the functor𝐹𝑤∘ 𝐹𝑤. In this manner, we obtain a functor𝐹𝑔: R𝑛,𝑘→ R𝑛,𝑘for each word𝑔 ∈ ⟨𝑎, 𝑏⟩. We proceed by checking that the relations𝑎2= 1 and 𝑏3= 1 hold. Since the braids 𝑎2and𝑏3are both full twists with some framing dependency, it suffices to check that this full twist is homotopy equivalent to the identity object in the categoryR𝑛,𝑘.
Consider what happens when the tangles𝑎2or𝑏3are glued onto𝐸𝑛,𝑘. As framed tangles, the full twist𝑇3 is isotopic to the tangle pictured on the right below:
≃ ≃ (𝑡12𝑘2𝑞𝑘)3 .
Using the relations of Section 2, we can move the small projectors to the top and absorb the top twists. Using the big projector𝑃3𝑛,𝑘at the bottom to absorb the middle twist leaves us with
(𝑡12𝑘2𝑞𝑘)3(𝑡12𝑘2𝑞𝑘)−1 = s
𝑘 ≃ .
The last equality follows because we have reduced the grading, see Section 2.4. We have shown that the relations𝑎2= 1 and 𝑏3= 1 hold up to homotopy.
The corollary below notes that the above theorem defines representations of the modular groupSL(2, ℤ) via pullback.
Corollary 4.6. The groupSL(2, ℤ) acts on the category R𝑛,𝑘.
Proof. To each element𝑥 ∈ SL(2, ℤ) associate the functor 𝐹𝜋(𝑥) defined in the proof of the theorem where 𝜋 : SL(2, ℤ) → PSL(2, ℤ).
4.1 Concluding remarks
Remark 4.7. The argument in Theorem 4.5 applies to the subcategoriesX𝑛= Kom∗1/2(𝑛, 𝑛, 𝑛, 𝑛) of Kom∗1/2(4𝑛),
see [8, Section 6]. The objects ofX𝑛are homotopy equivalent to convolutions of chain complexes of the form
D n
n n
n
(4.1)
where𝐷 is a disjoint union of intervals interfacing between four copies of the projector 𝑃𝑛. Reducing the grading so that the functor s𝑛acts by identity gives a new categoryK𝑛= X𝑛/s𝑛. After fixing one projector, act on the other three by𝑛-cabled strands as in the proof above. This is a PSL(2, ℤ) representation.
It is possible to argue that the action ofPSL(2, ℤ) on the category K𝑛⊗ 𝔽2is strong by using results from previous papers. This material has been omitted.
Remark 4.8. The objects of the categoryK𝑛form a basis for the vector space assigned to the four punctured sphere𝑆24bySU(2) Chern–Simons theory. The quotient map, 𝑇 → 𝑆24, described in Section 3.2, identifies the SU(2) skein space for 𝑆2
4with theSO(3) skein space for 𝑇2(both with level𝑟 = 2(𝑛 + 1)). See [1, 26].
Remark 4.9. Spin networks of the form (4.1) correspond toU𝑞sl(2)-equivariant maps from the 𝑛th irreducible representation𝑉𝑛to𝑉𝑛⊗3. The dimension of this space is𝑛 + 1. We may view the categories K𝑛as a sequence of representations of increasing complexity.
Remark 4.10. It might be interesting to study the groups𝐾0(K𝑛) and 𝐾0(R𝑛,𝑘). The groups 𝐾0(E𝑛,𝑘) and 𝐾0(X𝑛)
are determined in [7, 8], but these groups are not the same because the identity s𝑘≅ Id implies that there must be other relations.
Remark 4.11. Tensoring the decomposition of identity in equation (2.4) with a chain complex representing
the full twist and applying Property 2 shows that the projectors in Theorem 2.8 diagonalize the action of the centerZ(B𝑛) on Kom∗1/2(𝑛). In order to reduce an arbitrary 𝐺-action to a 𝐺/Z(𝐺)-action in a more general setting, one must first find an extension ofC in which the action of the center Z(𝐺) respects the grading. Some results predict such decompositions for other braid group representations related to knot homologies [13, Theorem 1.1].
References
[1] B. Bakalov and A. A. Kirillov, Lectures on Tensor Categories and Modular Functors, Univ. Lecture Ser. 21, American Mathematical Society, Providence, 2001.
[2] D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499. [3] D. Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007), no. 3, 243–255.
[4] J. Birman, Braids, Links and Mapping Class Groups, Ann. of Math. Stud. 82, Princeton University Press, Princeton, 1975. [5] A. I. Bondal and M. M. Kapranov, Enhanced triangulated categories, Math. USSR Sb. 70 (1991), no. 1, 93–107.
[6] W.-L. Chow, On the algebraic braid group, Ann. of Math. (2) 49 (1948), 654–658.
[7] B. Cooper and M. Hogancamp, An exceptional collection for Khovanov homology, preprint (2012), http://arxiv.org/abs/ 1209.1002.
[8] B. Cooper and V. Krushkal, Categorification of the Jones-Wenzl projectors, Quantum Topol. 3 (2012), no. 2, 139–180. [9] B. Cooper and V. Krushkal, Handle slides and localizations of categories, Int. Math. Res. Not. IMRN 2012 (2012),
DOI 10.1093/imrn/rns108.
[10] T. Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), no. 2, 223–274. [11] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton University Press, Princeton, 2011.
[12] I. Frenkel, C. Stroppel and J. Sussan, Categorifying fractional Euler characteristics, Jones–Wenzl projector and 3𝑗-symbols with applications to Exts of Harish-Chandra bimodules, Quantum Topol. 3 (2012), no. 2, 181–253.
[13] E. Gorsky and A. Negut, Refined knot invariants and Hilbert schemes, preprint (2012), http://arxiv.org/abs/1304.3328. [14] M. Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665–741.
[15] M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), no. 1, 203–271.
[16] R. Lipshitz, P. Ozváth and D. Thurston, A faithful linear-categorical action of the mapping class group of a surface with boundary, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1279–1307.
[17] S. MacLane, Categories for the Working Mathematician, Grad. Texts in Math. 5, Springer-Verlag, New York, 1998. [18] G. Moore and N. Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), no. 2, 177–254. [19] J. Roberts, Rozansky–Witten theory, in: Topology and Geometry of Manifolds (Athens 2001), Proc. Sympos. Pure Math. 71,
American Mathematical Society, Providence (2003), 1–17.
[20] R. Rouquier, Categorification of sl(2) and braid groups, in: Trends in Representation Theory and Related Topics, American Mathematical Society, Providence (2006), 137–167.
[21] L. Rozansky, An infinite torus braid yields a categorified Jones–Wenzl projector, preprint (2010), http://arxiv.org/abs/ 1005.3266.
[22] L. Rozansky, A categorification of the stable SU(2) Witten–Reshitikhin–Turaev invariant of links in 𝕊2× 𝕊1, preprint (2012),
http://arxiv.org/abs/1011.1958.
[23] L. Rozansky, Khovanov homology of a unicolored 𝐵-adequate link has a tail, preprint (2012), http://arxiv.org/abs/1203. 5741.
[24] P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108.
[25] B. Toën, Lectures on dg-categories, in: Topics in Algebraic and Topological 𝐾-Theory (Sedano 2007), Lecture Notes in Math. 2008, Springer-Verlag, Berlin (2011), 243–302.
[26] K. Walker, On Witten’s 3-manifold invariants, preprint (1991), http://canyon23.net/math/.
[27] B. Webster, Knot invariants and higher representation theory II, preprint (2013), http://arxiv.org/abs/1005.4559. Received December 15, 2013; revised May 16, 2014.
